LS-DYNA Beam Elements : Default and User Defined Cross Section Integration

LS-DYNA provides several beam element formulations, see the keyword description for *Section_Beam in the User’s Manual. Several of these beam element formulations support user supplied integration of the cross section, via the *Integration_Beam keyword. While most LS-DYNA users are familiar with the similar through-the-thickness integration algorithm for shell elements, which is made trivial by the rectangular cross section geometry assumed for shell elements, the numerical integration of even simple beam element cross sections requires more effort, and as will be demonstrated, more planning. In this article, a detailed explanation of the beam element cross section integration algorithm is presented. Simple suggestions for calculating, and checking, user provided integration rules are illustrated through several examples. The examples also provide suggestions for improving the LS-DYNA Standard Cross Section Types, available via the ICST parameter of the *Integration_Beam keyword. ANALYTICAL INTEGRATION To provide a basis for assessing the numerical integration algorithm, used in LSDYNA, for beam element cross sections, we first establish the corresponding analytical results. There are two cross section integrations that will be considered: axial force and bending moment; the former is relatively straight forward and is included only for completeness. Axial Force The axial force in a beam element is obtained by integrating the axial stress over the cross section, ( ) , F dA y z dydz σ σ = = ∫ ∫ (1) where ( ) , y z σ σ = is the axial stress which typically varies over the cross sectional area in the y z − plane. Bending Moment The moments, about the local y and z axes, in a beam element are obtained by integrating the axial stress times the appropriate distance (moment arm) over the cross section, ( ) ( ) , , y z M zdA y z zdydz M ydA y z ydydz σ σ σ σ = = = = ∫ ∫ ∫ ∫ (2) Implicit / New Developments 4 European LS-DYNA Users Conference